J Algebr Comb (2011) 33: 163–198
DOI 10.1007/s10801-010-0238-4
A computational and combinatorial exposé
of plethystic calculus
Nicholas A. Loehr · Jeffrey B. Remmel
Received: 29 January 2010 / Accepted: 19 May 2010 / Published online: 12 June 2010
© The Author(s) 2010. This article is published with open access at Springerlink.com
Abstract In recent years, plethystic calculus has emerged as a powerful technical
tool for studying symmetric polynomials. In particular, some striking recent advances
in the theory of Macdonald polynomials have relied heavily on plethystic computations. The main purpose of this article is to give a detailed explanation of a method
for finding combinatorial interpretations of many commonly occurring plethystic expressions, which utilizes expansions in terms of quasisymmetric functions. To aid
newcomers to plethysm, we also provide a self-contained exposition of the fundamental computational rules underlying plethystic calculus. Although these rules are
well-known, their proofs can be difficult to extract from the literature. Our treatment
emphasizes concrete calculations and the central role played by evaluation homomorphisms arising from the universal mapping property for polynomial rings.
Keywords Plethysm · Symmetric functions · Quasisymmetric functions · LLT
polynomials · Macdonald polynomials
1 Introduction
1.1 Plethysm
The plethysm F [G] of a symmetric polynomial F (x) with a symmetric polynomial
G(x) is essentially the polynomial obtained by substituting the monomials of G(x)
First author supported in part by National Security Agency grant H98230-08-1-0045.
N.A. Loehr ()
Virginia Tech, Blacksburg, VA 24061-0123, USA
e-mail: nloehr@vt.edu
J.B. Remmel
University of California, San Diego, La Jolla CA 92093-0112, USA
e-mail: jremmel@ucsd.edu
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for the variables of F (x). This operation was introduced by Littlewood [31] in the
study of group representation theory. This operation is often called “outer plethysm”
to distinguish it from the operation of “inner plethysm” or Kronecker product of representations. One of the major open problems in the theory of symmetric functions
and the representation theory of classical groups is to be able to compute the coeffiν in the expansion
cients aλ,μ
sλ [sμ ] =
ν
aλ,μ
sν (x)
(1)
ν
where sλ , sμ , sν denote the Schur functions corresponding to partitions λ, μ and ν,
respectively. We note that Littlewood used the notation {μ} ⊗ {λ} for sλ [sμ ]. The operation of plethysm arises naturally in both the representation theory of the general
linear group GL(n, C) and the symmetric group Sn . For example,
is a finite di if E mensional vector space over a field of characteristic 0, we let λ E and μ E denote
the representation space for the irreducible representation of GL(E) associated
with
ν gives the multiplicity of ν E in the
the partitions λ and μ. Then the coefficient aλ,μ
direct sum decomposition of λ ( μ E). Similarly, if λ is a partition of n and μ is
a partition of m, sλ [sμ ] can be viewed as the Frobenius image of the character of the
representation of Smn which can be described as follows. Let Aλ be the irreducible
Sn -module corresponding to λ, and let Aμ be the irreducible Sm -module corresponding to μ. The wreath product Sm Sn , which is the normalizer of Snm = Sn × · · · × Sn
in Smn , acts on Aλ and on the mth tensor power T m (Aμ ) and, hence, it also acts
on Aλ ⊗ T m (Aμ ). Then sλ [sμ ] is the Frobenius image of the character of the Smn module induced by Aλ ⊗ T m (Aμ ), see [34] or [27]. To date, there is no satisfactory
ν . There are only a very few special
combinatorial description of the coefficients aλ,μ
cases of λ and μ, such as Littlewood’s expansion [32] of s2 [sn ] and s(12 ) [sn ], where
ν . However, there are a variety of algorithms to
we have explicit formulas for aλ,μ
ν
compute aλ,μ , see [1, 8–10, 26, 35, 42].
While the representation-theoretic motivation for the operation of plethysm is
clear, one can more generally consider the operation of plethysm of any two symmetric functions F [G]. In particular, the plethysm operation can also be formulated
in the more abstract setting of λ-rings. The notion of a λ-ring was introduced by
Grothendieck [21] in the study of Chern classes. Later Atiyah [2] used λ-rings and
the theory of representations of Sn over the complex numbers C to investigate operations in K-theory. Atiyah and Tall [3] and Knutson [28] showed that the Grothendieck
representation ring R = R(Sn ) of the symmetric group Sn forms a special λ-ring with
respect to exterior power. In fact, the Hopf ring of symmetric functions in countably
many variables, see [19], is a free λ-ring on the elementary symmetric function e1 .
Thus the graded Hopf ring R is also a free λ-ring on one generator F −1 (e1 ) = η1
where F : R → H is the Frobenius map defined in [3, 28]. Thus η1 is the class of
the trivial representation. In fact, the λ-ring structure on R can be derived from the
plethysm operation, see [40]. This connection with λ-rings leads to an axiomatic presentation of plethysm as developed, for example, in [28].
Computations involving plethysm have many applications. For example, Hoffman [25] investigated inner and outer plethysms in the framework of τ -rings.
J Algebr Comb (2011) 33: 163–198
165
Wybourne [41] showed that there are many applications of plethysm in physics. The
special case of the expansion of the plethysm sm [sn ] has applications in nineteenthcentury invariant theory, see [11, 31]. Plethysm plays an important role in the theory of symmetric functions and Schubert polynomials. It can be used to unify many
proofs of old identities and has been used to prove a host of new results. This is
beautifully illustrated in Alain Lascoux’s book [29].
1.2 Plethystic calculus
The subject of this paper is plethystic calculus, which is an extension of the plethysm
operation to expressions of the form F [G], where F is a symmetric function (or
a formal limit of symmetric functions) and G is a formal power series or Laurent
series. This plethystic calculus has been used extensively by researchers including
Francois Bergeron, Nantel Bergeron, Adriano Garsia, Jim Haglund, Mark Haiman,
and Glenn Tesler (among others) in an ongoing study of the Bergeron–Garsia nabla
operator, diagonal harmonics modules, Macdonald polynomials, and related symmetric functions [4–6, 13–18, 22]. Plethystic calculus has become an indispensable
computational tool for organizing and manipulating intricate relationships between
symmetric functions.
Many commonly occurring symmetric functions (like skew Schur functions) have
well-known combinatorial interpretations involving tableaux or similar structures. In
contrast, it is not always easy to write down a combinatorial formula for a plethystic
expression F [G]. The primary goal of this paper is to give a detailed explanation of
a general technique for finding such combinatorial formulas for many choices of F
and G. This technique, which was used in [23, 24] to study Macdonald polynomials,
employs an extension of plethystic calculus in which F is allowed to be a quasisymmetric function. We will give a complete, self-contained description of this technique
in Sect. 4, filling in many details that are only hinted at in [23, 24].
The second goal of this paper is to give a rigorous, detailed account of the algebraic
foundations of the plethystic calculus using a minimum of technical machinery. We
will supply complete derivations of many plethystic identities that are well-known,
but whose proofs are difficult to extract from the literature. In particular, we will prove
the plethystic addition formula for skew Schur functions as well as plethystic versions
of the Cauchy identities. The starting point for our development of plethystic calculus
is the fact that the power-sums pn are algebraically independent elements that generate the ring of symmetric functions. Plethystic notation gives a concise way to define
homomorphisms on this ring by specifying their effect on every pn . It follows that
each plethystic identity ultimately results from properties of the transition matrices
between the power-sums and other bases of symmetric functions. We will see that
this leads to elementary computational or combinatorial proofs of many fundamental plethystic identities, which avoid the more technical aspects of λ-rings and Hopf
algebras. A slight disadvantage of this approach is that we must restrict ourselves to
working over fields of characteristic zero.
This paper is organized as follows. Section 2 develops the basic algebraic properties of plethysm from scratch, using a definition based on power-sum symmetric
functions and evaluation homomorphisms. Section 3 gives two proofs of the crucial addition formula for plethystic evaluation of skew Schur functions. Section 4
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describes a method for finding combinatorial interpretations of a variety of plethystic expressions. In particular, we show how plethystic transforms of quasisymmetric functions provide combinatorial interpretations for plethystic evaluations of skew
Schur functions and Lascoux–Leclerc–Thibon (LLT) polynomials. Section 5 derives
the plethystic Cauchy formulas and illustrates their use by giving an application (due
to Garsia) to the theory of Macdonald polynomials.
2 Plethysm and power-sums
Throughout this paper, let K denote a field of characteristic zero. This section defines the graded ring Λ of symmetric functions with coefficients in K, Littlewood’s
binary plethysm operation on Λ, and extended versions of this operation commonly
called “plethystic notation.” The theorems in this section are all well-known among
specialists, although our proofs are more detailed and less technical than those found
in the standard references [28, 34]. Our approach is based on the universal mapping
properties of polynomial rings (reviewed below) and accords a central role to the
power-sum symmetric functions.
2.1 Review of polynomial rings
Let K[z1 , . . . , zN ] denote the polynomial ring in N variables with coefficients in K.
This ring is a K-algebra satisfying the following universal mapping property (UMP):
for every1 K-algebra S and every N -tuple (a1 , . . . , aN ) of elements of S, there exists
a unique K-algebra homomorphism φ : K[z1 , . . . , zN ] → S such that φ(zi ) = ai for
1 ≤ i ≤ N . This homomorphism is given explicitly by
cβ z β =
cβ a β (cβ ∈ K),
φ
β∈NN
β
N βi
βi
β
where we write zβ = N
i=1 zi and a =
i=1 ai . For f ∈ K[z1 , . . . , zN ], we often write f (a1 , . . . , aN ) to denote the element φ(f ) ∈ S. We call φ the evaluation
homomorphism determined by setting zi = ai .
We will also need polynomial rings in countably many indeterminates. For
M < N , we can view K[z1 , . . . , zM ] as a subset of K[z1 , . . . , zN ] in the natural way.
Now define the set
∞
R = K {zi : i ≥ 1} =
K[z1 , . . . , zN ].
N =1
With the obvious definitions of addition and multiplication, R becomes a K-algebra.
Moreover, R satisfies the expected universal mapping property: for every K-algebra
S and indexed family {ai : i ≥ 1} ⊆ S, there exists a unique K-algebra homomorphism φ : R → S such that φ(zi ) = ai for all i ≥ 1. For more information on polynomial rings, see Chap. IV of [7].
1 We assume throughout that all K-algebras under consideration are commutative, associative, and have a
unit element. All K-algebra homomorphisms are assumed to preserve the unit element.
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2.2 Symmetric polynomials, symmetric functions, and power-sums
Consider the polynomial ring RN = K[x1 , . . . , xN ]. By the universal mapping property, every permutation w ∈ SN induces a unique K-algebra endomorphism φw :
RN → RN such that φw (xi ) = xw(i) . By the uniqueness part of the UMP, we have
φv◦w = φv ◦ φw , because both sides are K-algebra homomorphisms on RN that send
xi to xv(w(i)) for 1 ≤ i ≤ N . Thus, SN acts on RN by “permuting the variables.”
Define
ΛN = f ∈ RN : φw (f ) = f for all w ∈ SN ,
which is a K-subalgebra of RN . Elements of ΛN are called symmetric polynomials in
N variables. RN and ΛN become graded algebras by letting each xi have degree 1.
The kth power-sum symmetric polynomial in N variables is pk,N = x1k + x2k +
k . For an integer partition μ = (μ ≥ μ ≥ · · · ≥ μ > 0), define p
·
· · + xN
1
2
s
μ,N =
s
p
.
Let
Par
be
the
set
of
all
integer
partitions,
and
let
Par(n)
be
the
set
j =1 μj ,N
of partitions of n. It is well-known that, for all N ≥ n, the indexed set {pμ,N : μ ∈
Par(n)} is a K-basis for the subspace ΛnN consisting of all polynomials in ΛN which
are homogeneous of degree n [38, 39].
Next we define the ring of abstract symmetric functions with coefficients in K to
be the polynomial ring
Λ = K {pi : i ≥ 1} ,
where the pi ’s are indeterminates called abstract power-sum symmetric functions.
We
define a grading on Λ by letting deg(pi ) = i. For every partition μ, define pμ =
i≥1 pμi . It follows from the very definition of polynomial rings that the set {pμ :
μ ∈ Par(n)} is a K-basis for the subspace Λn consisting of all polynomials which
are homogeneous of
degree n, and the set {pi : i ≥ 1} is algebraically independent
over K. Note Λ = n≥0 Λn .
To relate abstract symmetric functions to concrete symmetric polynomials, introduce the evaluation homomorphisms evN : Λ → RN such that evN (pi ) = pi,N =
i . For f ∈ Λ, we will often write f (x , . . . , x ) instead of ev (f ). One
x1i + · · · + xN
1
N
N
can show that for N ≥ n, evN restricts to a vector space isomorphism of Λn onto ΛnN .
2.3 Littlewood’s plethysm operation on Λ
We are about to define a binary operation • : Λ × Λ → Λ called plethysm, first introduced by Littlewood [32]. For f, g ∈ Λ, f • g (also denoted f [g]) is called the
plethystic substitution of g into f . (Some authors write f ◦ g instead of f • g. We
reserve the open circle ◦ to denote composition of functions.) The results presented
next are equivalent to those stated in [34, Sect. I.8], but we start from a different
definition of plethysm that avoids the use of “fictitious variables.”
We want the plethysm operation to satisfy the following three basic properties.
P1. For all m, n ≥ 1, pm • pn = pmn .
P2. For all m ≥ 1, let Lm : Λ → Λ be the map Lm (g) = pm • g (“left plethysm
by pm ”). Then Lm is a K-algebra homomorphism.
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P3. For all g ∈ Λ, let Rg : Λ → Λ be the map Rg (f ) = f •g (“right plethysm by g”).
Then Rg is a K-algebra homomorphism.
Spelled out in more detail, property P2 says that for all m ≥ 1, g1 , g2 ∈ Λ, and
c ∈ K,
pm • (g1 + g2 ) = pm • g1 + pm • g2 ,
pm • (g1 · g2 ) = (pm • g1 ) · (pm • g2 ),
pm • c = c.
Property P3 says that for each g, f1 , f2 ∈ Λ and c ∈ K,
(f1 + f2 ) • g = f1 • g + f2 • g,
(f1 · f2 ) • g = (f1 • g) · (f2 • g),
c • g = c.
Theorem 1 There exists a unique binary operation • on Λ satisfying P1, P2, and P3.
Proof Fix m ≥ 1. By the UMP for polynomial rings, there is a unique K-algebra
homomorphism Lm : Λ → Λ such that Lm (pn ) = pmn for all n. So, for each fixed
m ≥ 1, there is a unique way of defining pm • g (g ∈ Λ) so that P1 and P2 hold.
Now fix g ∈ Λ and let m vary. Using the UMP again, we see that there is a unique
K-algebra homomorphism Rg : Λ → Λ such that Rg (pm ) = pm • g. So there is a
unique definition of f • g (for f ∈ Λ) satisfying P3.
To compute
f • g in practice, first express f in terms of the power-sum basis of Λ,
say f = ν cν pν . Then, thanks to P3,
f •g=
ν
cν
(pνi • g).
i
Second, write g in terms of the power-sum basis, say g = μ dμ pμ . Then, by P2
and P1,
pν i • g =
dμ (pνi • pμj ) =
dμ (pνi μj ).
μ
j
μ
j
We will see that the most basic plethystic identities all follow from the universal
mapping properties for polynomial rings.
Theorem 2 (Plethysm vs. Substitution) For all g ∈ Λ and m ≥ 1,
m
.
(pm • g)(x1 , . . . , xN ) = g x1m , . . . , xN
Proof Define evaluation homomorphisms evN : Λ → K[x1 , . . . , xN ] and πm :
k and π (x ) =
K[x1 , . . . , xN ] → K[x1 , . . . , xN ] by setting evN (pk ) = x1k + · · · + xN
m i
m
xi for all k, i ≥ 1. The theorem asserts that evN ◦Lpm = πm ◦ evN . This is true
because both sides are K-algebra homomorphisms with domain Λ that send pk to
mk .
x1mk + · · · + xN
Theorem 3 (Centrality of pn ) For all h ∈ Λ and all n ≥ 1, pn • h = h • pn .
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Proof Consider the K-algebra homomorphisms Ln and Rpn on Λ. For all m ≥ 1,
property P1 shows that
Ln (pm ) = pn • pm = pnm = pmn = pm • pn = Rpn (pm ).
By the uniqueness part of the UMP for polynomial rings, Ln = Rpn . Applying these
functions to h gives the result.
Theorem 4 (Unit Element for Plethysm) For all h ∈ Λ, p1 • h = h = h • p1 .
Proof By the last proposition with n = 1, we need only prove the first equality. By
property P2, L1 is the unique K-algebra homomorphism on Λ sending pm to p1 •
pm = pm for all m. Since the identity map on Λ also sends pm to pm for all m,
uniqueness shows that L1 = idΛ . So p1 • h = id(h) = h.
Theorem 5 (Associativity of Plethysm) For all f, g, h ∈ Λ, (f • g) • h = f • (g • h).
Proof Step 1: The result holds when f = pi , g = pj , and h = pk . For in this case,
repeated use of property P1 shows that both sides equal pij k .
Step 2: The result holds for f = pi , g = pj , and all h ∈ Λ. Since f • g = pij , we
must prove that Lij = Li ◦ Lj . This holds since both sides are K-algebra homomorphisms of Λ that agree on all pk ’s (by step 1).
Step 3: The result holds for f = pi and all g, h ∈ Λ. Here we must prove that
Rh ◦ Li = Li ◦ Rh . By step 2, both sides are K-algebra homomorphisms of Λ having
the same effect on every pj . So they are equal by the UMP for Λ.
Step 4: The result holds for all f, g, h ∈ Λ. In this final step, we must show that
Rh ◦ Rg = Rg•h . This holds since both sides are K-algebra homomorphisms of Λ
that agree on all pi ’s (by step 3).
2.4 Plethystic notation
Classically, the plethysm f • g was only defined when f and g both belong to Λ.
Researchers including Francois Bergeron, Nantel Bergeron, Adriano Garsia, Jim
Haglund, Mark Haiman, and Glenn Tesler (among others) extended the idea of
plethystic substitution to situations where f ∈ Λ and g belongs to some larger
K-algebra. This led to the development of a plethystic calculus that has become an
enormously useful computational tool for proving results about Macdonald polynomials, the Bergeron–Garsia nabla operator, and related constructs [4–6, 13–18, 22].
We can define this extended version of plethysm by suitably modifying the axioms
P1, P2, and P3. We take as initial data a K-algebra Z and, for each integer m ≥ 1,
a Q-algebra homomorphism Lm : Z → Z. (In many applications, Z contains Λ as a
subalgebra, and Lm |Λ is the usual map pn → pmn .) For every g ∈ Z, write Lm (g) =
pm • g. By the UMP for Λ, we have for each g ∈ Z a K-algebra homomorphism
Rg : Λ → Z such that Rg (pm ) = pm • g. We now have an operation
·[·] : Λ × Z → Z
given by f [g] = Rg (f ).
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(Here we have reverted to the notation for plethysm now in common use, in which the
second argument is enclosed by square brackets.) When f = pm , we have pm [g] =
Rg (pm ) = pm • g = Lm (g). To compute f [g] for a general f ∈ Λ, express f as a Klinear combination of products of pm ’s and then replace each pm by pm [g] = Lm (g).
The following example illustrates a typical application of this general setup.
Example 1 Let K be the field Q(q, t), and let z, w, y be some additional variables.
Let Z = Λ(z, w, y) be the fraction field of the polynomial ring Λ[z, w, y], so that
Z∼
= Q[p1 , . . . , pn , . . .](q, t, z, w, y).
Using the UMP’s for polynomial rings and fraction fields, we see that for each m ≥ 1,
there is a unique Q-algebra endomorphism of Z such that pn → pnm for all n,
q → q m , t → t m , z → zm , w → w m , and y → y m . Informally, this means that we
compute pm [g] by replacing every “variable” in g by its mth power (cf. Theorem 2).
In this informal description, we are viewing q, t, z, w, y as “variables” (even though
q, t ∈ K), and we think of pn as an infinite sum i≥0 xin . Then the rule pn → pnm
arises by replacing each “variable” xi by its mth power. To compute f [g] for arbitrary f ∈ Λ, express f as a sum of products of power-sums and use the previous rule.
Note that Lm is not a K-algebra homomorphism for m > 1, since Lm (q) = q m .
Remark 1 Researchers are continually extending plethystic notation to cover successively more general situations. The key point to remember is that a plethystic expression of the form f [A] always denotes the image of f ∈ Λ under some K-algebra
homomorphism φA of Λ, where φA is supposed to be determined in some natural
way by the “plethystic alphabet” A. The following list gives some conventions that
have been developed for converting certain alphabets A to the associated homomorphism φA .
– Writing X = x1 + x2 + · · · + xN + · · · , we have by convention f [X] = f (so
plethystic substitution of X into f is just the identity homomorphism on Λ). This
notation arises by analogy with the finite version f [x1 + · · · + xN ] = evN (f ) =
f (x1 , . . . , xN ). Instead of f [X], it would be more precise to write f [p1 ] (cf. Theorem 4).
– The ring Λ is also a Hopf algebra with a comultiplication map Δ : Λ → Λ ⊗K Λ.
This map is a K-algebra homomorphism defined (using the UMP) by setting
Δ(pk ) = pk ⊗ 1 + 1 ⊗ pk for all k ≥ 1. By convention, the plethystic notation f [X + Y ] denotes Δ(f ). (Thus f [X + Y ] is really an abbreviation for
f [p1 ⊗ 1 + 1 ⊗ p1 ].) We will obtain a formula for Δ(sλ/ν ) in the next section.
– Let ω : Λ → Λ be the usual involutory K-automorphism of Λ defined (via the
UMP) by sending each pk to (−1)k−1 pk . The plethystic expressions f [−− X] =
f [−X] are used by some authors to stand for ω(f ). By contrast, f [−X] =
f [−p1 ] is the image of f under the homomorphism sending each pk to −pk (the
antipode map for the Hopf algebra Λ).
Theorem 6 (Negation Rule) If g ∈ Λn is homogeneous of degree n and A is any
plethystic alphabet, then
g[−A] = (−1)n ω(g) [A].
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171
Proof Let φA be the K-algebra homomorphism with domain Λ determined by A.
Since Lm is assumed to be a group homomorphism, we have pm [−A] = −pm [A] =
(−1)m (−1)m−1 pm [A] = (−1)m (ω(pm ))[A] for all m. Next, if μ = (μ1 , . . . , μs ) is a
partition of n,
pμ [−A] =
s
pμi [−A] =
i=1
s
(−1)μi ω(pμi ) [A]
i=1
= (−1)n
s
φA ω(pμi ) = (−1)n ω(pμ ) [A],
(2)
i=1
where the last step followssince φA and ω are ring homomorphisms. Finally, given
g ∈ Λn , we can write g = μ∈Par(n) cμ pμ for suitable cμ ∈ K. Then, by K-linearity,
g[−A] =
cμ pμ [−A] =
μ
= (−1) ω
n
cμ (−1)n ω(pμ ) [A]
μ
cμ pμ [A] = (−1)n ω(g) [A].
μ
(3)
Theorem 7 (Monomial Substitution Rule) In the context of Example 1, suppose A is
a finite sum of monic monomials M1 , . . . , MN in Z. For any g ∈ Λ,
g[A] = g(M1 , M2 , . . . , MN ),
where the right side denotes the image of g(x1 , . . . , xN ) = evN (g) under the evaluation homomorphism E : K[x1 , . . . , xN ] → Z that sends xi to Mi for all i.
Proof This result follows immediately from the UMP, since φA and E ◦ evN are
K-algebra homomorphisms that have the same effect on every pk .
Example 2 For any symmetric function f ,
f 2q + qt + 3t 4 = f q + q + qt + t 4 + t 4 + t 4 = f q, q, qt, t 4 , t 4 , t 4 .
As another example, note that
2
p(2,2) [3t] = p(2,2) (t, t, t) = p2 (t, t, t)2 = t 2 + t 2 + t 2 = 9t 4 .
However, evaluating p(2,2) (x1 ) at x1 = 3t gives
2
p(2,2) (3t) = p2 (3t)2 = (3t)2 = 81t 4 = p(2,2) [3t].
This example shows that the monomials involved in the last theorem must be monic.
It also shows that one must take care to distinguish the square plethystic brackets
from ordinary round parentheses used to denote the image of a polynomial under an
evaluation homomorphism.
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Example 3 Let f ∈ Λ and μ ∈ Par(n). The bi-exponent generator of μ is
Bμ =
q i t j ∈ Q(q, t).
(i,j ):1≤j ≤μi
Let (i1 , j1 ), . . . , (in , jn ) be any ordering of the pairs (i, j ) appearing in this sum.
Treating q and t as variables, we see that f [Bμ ] can be computed by evaluating f (x1 , . . . , xn ) at xk = q ik t jk . This gives a concrete way of thinking about the
plethysm f [Bμ ], which occurs frequently in the theory of Macdonald polynomials.
3 Plethystic calculus and Schur functions
This section discusses the plethystic addition formula for simplifying sλ/ν [A + B],
where sλ/ν denotes a skew Schur function. This addition formula is known from the
theory of λ-rings (cf. [34, pp. 72, 74]), but we will provide two elementary proofs that
make no use of λ-rings. We begin by reviewing the relevant combinatorial definitions.
3.1 Review of Schur functions
Suppose ν = (νi : i ≥ 1) and λ = (λi : i ≥ 1) are integer partitions such that ν ⊆ λ,
i.e., νi ≤ λi for all i. The skew diagram λ/ν is the set of all points (i, j ) ∈ N+ ×
N+ satisfying νi < j ≤ λi . A semistandard tableau of shape λ/ν over the alphabet
[N ] = {1, 2, . . . , N} is a function T : λ/ν → [N ] that weakly increases along rows
and strictly increases up columns. More precisely, this means T (i, j ) ≤ T (i, j + 1)
whenever (i, j ) and (i, j + 1) both lie in λ/ν; and T (i, j ) < T (i + 1, j ) whenever
(i, j ) and (i + 1, j ) both lie in λ/ν. We write SSYTN (λ/ν) for the
set of all such
tableaux. The weight monomial of such a tableau is wt(T ) = x T = (i,j )∈λ/ν xT (i,j ) .
The skew Schur polynomial in N variables indexed by λ/ν is
sλ/ν,N =
x T ∈ K[x1 , . . . , xN ].
(4)
T ∈SSYTN (λ/ν)
One can prove that, for N ≥ n, the set {sλ/0,N : λ ∈ Par(n)} is a basis for the K-vector
space ΛnN [38, 39].
Next we describe the expansion of skew Schur polynomials in terms of power-sum
symmetric polynomials. For this we need a few more definitions. Suppose
μ is a partition of n with ai parts equal to i, for 1 ≤ i ≤ s. Then zμ is the integer si=1 (i ai ai !).
We note that n!/zμ is the number of permutations of n objects with cycle type μ.
A skew shape is a ribbon iff it consists of a connected sequence of squares that contains no 2 × 2 rectangle. A k-ribbon is a ribbon consisting of k squares. The spin,
spin(S), of a ribbon S that occupies j rows is defined by spin(S) = j − 1 and the
sign, sgn(S), of S is defined by sgn(S) = (−1)j −1 . For example, the skew shape
(5, 5, 2, 2, 1)/(4, 1, 1), whose diagram appears below, is a 9-ribbon of spin 4 and
J Algebr Comb (2011) 33: 163–198
173
sign +1.
Now suppose λ/ν is a skew shape with n squares and α is a composition of n.
A rim hook tableau of shape λ/ν and type α is a sequence of partitions T = (ν 0 ⊆
ν 1 ⊆ · · · ) such that ν 0 = ν, ν N = λ for all sufficiently large N , and ν i /ν i−1 is an
αi -ribbon for all i ≥ 1. T is most easily visualized by placing an i in each square of
the αi -ribbon ν i /ν i−1 . For example, the following picture represents the rim hook
tableau
T = (2, 2, 1), (5, 3, 1), (5, 4, 4, 2), (5, 4, 4, 4), (5, 4, 4, 4), (5, 5, 5, 5), (5, 5, 5, 5, 1)
of shape (5, 5, 5, 5, 1)/(2, 2, 1) and type (4, 6, 2, 0, 3, 1).
6
2 2 3 3
2 2 2
1 2
1 1
5
5
5
1
We define spin(T ) = i spin(ν i /ν i−1 ) and sgn(T ) = (−1)spin(T ) . Our example has
λ/ν
spin 5 and sign −1. Define χα = T sgn(T ), where we sum over all rim hook
tableaux T of shape λ/ν and type α. It can be shown that reordering the parts of α
λ/ν
does not change χα . Furthermore, we have the following formula for skew Schur
polynomials:
−1 λ/ν
N ≥ n = |λ/ν| .
(5)
zμ
χμ pμ,N
sλ/ν,N =
μ∈Par(n)
Keeping in mind the isomorphism ΛnN ∼
= Λn (for N ≥ n), we now define abstract
skew Schur functions by setting
−1 λ/ν
zμ
χμ pμ ∈ Λ.
(6)
sλ/ν =
μ∈Par(n)
As special cases of this definition, we obtain the complete symmetric functions hn =
symmetric functions en = s(1n )/0 . Furthermore, we set h0 =
s(n)/0 and the elementary
e0 = 1, hμ = i≥1 hμi , and eμ = i≥1 eμi (these are also special cases of skew
Schur functions). It can be shown that, for n ≥ 0, {eμ : μ ∈ Par(n)} and {hμ : μ ∈
Par(n)} are both bases for the K-vector space Λn . This is equivalent to the fact that
{en : n ≥ 1} and {hn : n ≥ 1} are algebraically independent over K. Thus, we can
view the ring Λ as a polynomial ring in three different ways:
Λ = K[p1 , p2 , . . .] ∼
= K[e1 , e2 , . . .] ∼
= K[h1 , h2 , . . .].
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Example 4 Let us use (6) to show that ω(sλ/ν ) = sλ /ν , where the prime denotes
conjugation. First, if T is a rim hook tableau of shape λ/ν, then the conjugate T of
T is a rim hook tableau of shape λ /ν . For example, if T is the rim hook tableau
pictured above, then T is pictured below.
1 5 5 5
1 2 2 3
1 1 2 3
2 2
2 6
Moreover, if R is a ribbon, then one easily checks that spin(R ) = |R| − 1 − spin(R)
and sgn(R ) = (−1)|R|−1 sgn(R). Hence if T is a rim hook tableau of shape λ/ν
and type μ ∈ Par, then T is a rim hook tableau of shape λ /ν and type μ such
λ /ν that sgn(T ) = (−1)|λ/ν|−(μ) sgn(T ), where (μ) is the length of μ. Thus χμ =
λ/ν
(−1)|λ/ν|−(μ) χμ and
ω(sλ/ν ) =
−1 λ/ν
zμ
χμ ω(pμ )
μ∈Par(n)
=
−1 λ/ν
zμ
χμ (−1)|λ/ν|−(μ) pμ
μ∈Par(n)
=
−1 λ /ν
zμ
χμ pμ
μ∈Par(n)
= sλ /ν .
By the negation rule, we therefore have
sλ/ν [−A] = (−1)|λ/ν| sλ /ν [A].
In particular, for μ ∈ Par(n), eμ [−A] = (−1)n hμ [A] and hμ [−A] = (−1)n eμ [A].
3.2 Plethystic addition formula
The “plethystic addition formula” is usually written
sλ/ν [A + B] =
sμ/ν [A]sλ/μ [B].
μ:ν⊆μ⊆λ
To emphasize the role played by homomorphisms and the UMP, we will rewrite this
formula in the following somewhat more general form. Suppose D and E are any
K-algebra homomorphisms from Λ into a K-algebra S. We write D +P E to denote
the unique K-algebra homomorphism from Λ to S that sends pk to D(pk ) + E(pk )
for all k ≥ 1.
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175
Theorem 8 For any skew shape λ/ν of size n, we have
(D +P E)(sλ/ν ) =
D(sμ/ν )E(sλ/μ ).
(7)
μ∈Par
ν⊆μ⊆λ
We will now give a new computational proof of this theorem based on the combinatorial formula (6). A different, more abstract proof starting from (4) is given in
Sect. 3.4. Expanding sλ/ν in terms of power-sum symmetric functions and then using
the definition of the homomorphism D +P E, we have
(D +P E)(sλ/ν ) =
γ ∈Par(n)
λ/ν (γ )
χγ D(pγi ) + E(pγi ) .
zγ
(8)
i=1
We proceed to prove some lemmas analyzing different components of this formula.
First some definitions: if α and β are compositions (or partitions) with (α) = s
and (β) = t, the concatenation α|β is the composition (α1 , . . . , αs , β1 , . . . , βt ). For
any composition α, let α + be the partition obtained by sorting the parts of α into
decreasing order.
Lemma 1 For γ ∈ Par,
(γ )
1 D(pγi ) + E(pγi ) =
zγ
i=1
α,β∈Par
(α|β)+ =γ
D(pα ) E(pβ )
·
.
zα
zβ
(9)
Proof Let γ have cj parts equal to j , for 1 ≤ j ≤ n. The left side of (9) can then be
written
n
c
1
D(pj ) + E(pj ) j .
c
c
c
1 1 2 2 · · · n n c1 !c2 ! · · · cn !
(10)
j =1
Expanding each factor (D(pj ) + E(pj
terms, this becomes
))cj
by the binomial theorem and rearranging
cj
n E(pj )cj −aj
D(pj )aj
·
aj !j aj
(cj − aj )!j cj −aj
j =1 aj =0
=
c1
a1 =0
···
cn n
n
D(pj )aj E(pj )cj −aj
.
aj !j aj
(cj − aj )!j cj −aj
an =0 j =1
(11)
j =1
Introduce new summation variables α, β ∈ Par by letting α have aj parts equal to j
and β have bj = cj − aj parts equal to j for all j . The multiple sum over a1 , . . . , an
becomes a sum over all partitions α and β such that (α|β)+ = γ . With this change of
variables, the right side of (11) becomes the right side of (9), as desired.
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Lemma 2 Suppose α is a composition of k, β is a composition of m, γ = α|β, and
λ/ν is a skew shape of size k + m. Then
λ/μ
χγλ/ν =
χαμ/ν χβ .
(12)
μ∈Par
ν⊆μ⊆λ,|μ/ν|=k
Proof A typical signed object counted by the left side looks like
T = ν 0 ⊆ ν 1 ⊆ · · · ⊆ ν s ⊆ · · · ⊆ ν s+t
where ν 0 = ν, ν s+t = λ, and ν i /ν i−1 is a γi -ribbon. Map T to the triple (μ, T 1 , T 2 ),
where μ = ν s , T 1 = (ν 0 ⊆ · · · ⊆ ν s ), and T 2 = (ν s ⊆ · · · ⊆ ν s+t ). This gives a bijection onto the set of signed objects enumerated by the right side of (12). Signs are
preserved, since spin(T ) = spin(T 1 ) + spin(T 2 ), so the lemma follows.
λ/ν
Since sorting the parts of γ does not change χγ , we conclude that
λ/ν
λ/μ
χαμ/ν χβ .
(13)
χ(α|β)+ =
μ∈Par
ν⊆μ⊆λ,|μ/ν|=|α|
Now we continue the proof of the main theorem. From (8) and (9), we get
D(pα ) E(pβ )
(D +P E)(sλ/ν ) =
χγλ/ν
·
.
zα
zβ
γ ∈Par(n)
λ/ν
Moving χγ
(14)
α,β∈Par
(α|β)+ =γ
inside the sum and using (13), this becomes
λ/μ D(pα ) E(pβ )
χαμ/ν χβ
·
.
zα
zβ
(15)
γ ∈Par(n) α,β∈Par μ∈Par
(α|β)+ =γ ν⊆μ⊆λ
|μ/ν|=|α|
Changing the order of summation (which allows us to eliminate the summation variable γ altogether) and regrouping, we obtain
λ/μ D(pα ) E(pβ )
χαμ/ν χβ
·
zα
zβ
μ∈Par α∈Par(|μ/ν|) β∈Par(|λ/μ|)
ν⊆μ⊆λ
=
μ∈Par α∈Par(|μ/ν|)
ν⊆μ⊆λ
μ/ν
χα D(pα )
zα
β∈Par(|λ/μ|)
λ/μ
χβ
E(pβ )
zβ
.
(16)
We now recognize the definitions of D(sμ/ν ) and E(sλ/μ ) appearing in this formula.
Thus we finally get the desired result
(D +P E)(sλ/ν ) =
D(sμ/ν )E(sλ/μ ).
μ∈Par
ν⊆μ⊆λ
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177
3.3 Consequences of the addition formula
Recalling that hm = s(m)/0 = s(m+k)/(k) and em = s(1m )/0 = s(1m+k )/(1k ) for all m and
k, we deduce the following formulas.
(D +P E)(hn ) =
n
(D +P E)(en ) =
D(hk )E(hn−k ),
k=0
n
D(ek )E(en−k ).
k=0
In plethystic notation, our formulas read:
sλ/ν [A + B] =
sμ/ν [A]sλ/μ [B],
μ:ν⊆μ⊆λ
hn [A + B] =
n
hk [A]hn−k [B],
k=0
en [A + B] =
n
ek [A]en−k [B].
k=0
Combining the addition and negation formulas, we also obtain the subtraction formulas:
sλ/ν [A − B] =
(−1)|λ/μ| sμ/ν [A]sλ /μ [B],
μ:ν⊆μ⊆λ
hn [A − B] =
n
(−1)n−k hk [A]en−k [B],
k=0
en [A − B] =
n
(−1)n−k ek [A]hn−k [B].
k=0
Example 5 Using our rules, we can now interpret more plethystic expressions as
encoding some rather unusual substitutions of variables. For instance,
e4 [2q + z − 3t − w] =
4
(−1)4−k ek (q, q, z)h4−k (t, t, t, w);
k=0
here we are using the subtraction rule with A = 2q + z = q + q + z and B = 3t + w =
t + t + t + w. Similarly,
h4 [x1 + x2 − y1 − y2 ] =
4
(−1)4−k hk (x1 , x2 )e4−k (y1 , y2 ).
k=0
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Example 6 Let us compute sν [1 − u], for ν ∈ Par. First, assume ν = (a, 1n−a ) is a
hook partition. The subtraction formula and Theorem 7 give
sν [1 − u] =
(−1)n−|μ| sμ (x1 )x →1 · sν /μ (x1 )x →u .
1
1
μ⊆ν
By considering semistandard tableaux, one easily sees that sρ/ξ (x1 ) is zero unless
|ρ/ξ |
ρ/ξ is a horizontal strip, in which case sρ/ξ (x1 ) = x1 . Therefore, in the preceding
formula, we only get a nonzero summand if both μ and ν /μ are horizontal strips.
This occurs iff μ = (a) or μ = (a − 1). Making the indicated substitutions for x1 , we
conclude that
s(a,1n−a ) [1 − u] = (−1)n−a un−a + (−1)n−a+1 un−a+1 = (−u)n−a (1 − u).
If ν ∈ Par is not a hook shape, one easily sees that for all μ ⊆ ν, one of μ or ν /μ
is not a horizontal strip. Thus, sν [1 − u] = 0 for all such ν. The calculations in this
example will be generalized in Sect. 4 below.
As another corollary of our addition formula, note that the comultiplication map
Δ has the form D +P E where D, E : Λ → Λ ⊗K Λ are given by D(pk ) = pk ⊗ 1
and E(pk ) = 1 ⊗ pk for all k. Therefore,
sμ/ν ⊗ sλ/μ .
(17)
Δ(sλ/ν ) =
μ:ν⊆μ⊆λ
In fact, we can use the comultiplication map Δ to give an alternate proof of Theorem 8.
3.4 Alternate proof of Theorem 8
The following proof is essentially a translation into the language of Hopf algebras of
the classical λ-ring approach to the plethystic addition formula.
Step 1. If λ/ν has size n and M, N ≥ n, then
sλ/ν (x1 , . . . , xM+N ) =
sμ/ν (x1 , . . . , xM )sλ/μ (xM+1 , . . . , xM+N ).
μ:ν⊆μ⊆λ
This identity is combinatorially evident, since the occurrences of 1, . . . , M in a
tableau T ∈ SSYTM+N (λ/ν) form a tableau of some shape μ/ν, whereas the remaining entries in T must then constitute a tableau of shape λ/μ.
Step 2. We prove the comultiplication formula (17). Consider the diagram of Kalgebra homomorphisms
Δ
Λ
evM ⊗ evN
evM+N
π
RM+N
Λ ⊗K Λ
RM ⊗ K RN
J Algebr Comb (2011) 33: 163–198
179
where RM = K[x1 , . . . , xM ], evM sends every pk to pk,M ∈ RM , RM+N and
= K[x
ev
are defined analogously, RN
M+1 , . . . , xM+N ], evN sends pk to
M+N
k
M<i≤M+N xi , and π is the canonical isomorphism sending f ⊗ g to f g. By checking on the algebra generators pk of Λ, we see that the diagram commutes. Step 1 says
that
sμ/ν ⊗ sλ/μ .
evM+N (sλ/ν ) = π evM ⊗ evN
μ:ν⊆μ⊆λ
On the other hand, commutativity of the diagram means that
evM+N (sλ/ν ) = π evM ⊗ evN Δ(sλ/ν ) .
Comparing these equations and noting that π is an isomorphism, we get
evM ⊗ evN
sμ/ν ⊗ sλ/μ = evM ⊗ evN Δ(sλ/ν ) .
μ:ν⊆μ⊆λ
The desired formula now follows since the restriction of evM ⊗ evN to the graded
component of Λ ⊗K Λ of degree n = |λ/ν| is a vector space isomorphism for
M, N ≥ n.
Step 3. With notation as in Theorem 8, consider the diagram of K-algebra homomorphisms
Δ
Λ
Λ ⊗K Λ
D+P E
D⊗E
mS
S
S ⊗K S
where mS is the map sending f ⊗ g to f g (this is a K-algebra homomorphism because S is commutative). This diagram commutes, as we see by checking on powersums. It follows that
(D +P E)(sλ/ν ) = mS ◦ (D ⊗ E) ◦ Δ(sλ/ν )
(by Step 2)
sμ/ν ⊗ sλ/μ
= mS ◦ (D ⊗ E)
= mS
μ:ν⊆μ⊆λ
D(sμ/ν ) ⊗ E(sλ/μ )
μ:ν⊆μ⊆λ
=
μ:ν⊆μ⊆λ
D(sμ/ν )E(sλ/μ ).
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4 Combinatorial interpretations of plethystic expressions
This section presents a new general technique for finding interpretations of plethystic expressions as sums of signed, weighted combinatorial objects. This technique
extends and systematizes a special case used in [23, 24] to provide combinatorial
formulas for certain plethystic transformations of Macdonald polynomials. The key
idea involves extending the plethystic calculus to apply to quasisymmetric functions,
then using standardization bijections to find quasisymmetric function expansions for
the symmetric functions under consideration. We begin by reviewing the necessary
facts concerning quasisymmetric functions, standard tableaux, and standardization in
Sects. 4.1, 4.2, and 4.3.
4.1 Quasisymmetric functions
For each n ≥ 1, the space Qn of quasisymmetric functions of degree n over K is defined to be a K-vector space of dimension 2n−1 with basis consisting of the symbols
Ln,S as S ranges over all subsets of {1, 2, . . . , n − 1}. Ln,S is called a fundamental
quasisymmetric function. By the universal mapping property for the basis of a vector
space, any function f mapping this basis into a K-vector space W uniquely extends
n
by linearity
nto a K-linear transformation Tf : Q → W . We remark that the space
Q = n Q can be made into a graded ring and a Hopf algebra, but we will only
need the vector space structure for our purposes here.
We can use the universal mapping property to define fundamental quasisymmetric
polynomials in N variables. For each N ≥ n, define an injective linear map evN,n
from Qn to K[x1 , . . . , xN ] by mapping Ln,S to
Ln,S (x1 , . . . , xN ) =
xi1 xi2 · · · xin ∈ K[x1 , . . . , xN ]
1≤i1 ≤i2 ≤···≤in ≤N
/
ik =ik+1 ⇒k ∈S
and extending by linearity (cf. [20, 38, 39, Chap. 7]). These polynomials “interpolate”
between the homogeneous and elementary symmetric polynomials, in the sense that
Ln,∅ (x1 , . . . , xN ) = hn (x1 , . . . , xN )
and
Ln,{1,2,...,n−1} (x1 , . . . , xN ) = en (x1 , . . . , xN ).
The image evN,n (Qn ) is denoted QnN and consists of the quasisymmetric polynomials
in N variables that are homogeneous of degree n.
It will be useful to express the formula for Ln,S (x1 , . . . , xN ) as an explicit sum of
weighted combinatorial objects. Let W (n, S, N) denote the set of all weakly increasing sequences w = w1 w2 · · · w
n with each wk ∈ {1, 2, . . . , N}, such that wk = wk+1
implies k ∈
/ S. Write wt(w) = ni=1 xw(i) . Then
Ln,S (x1 , . . . , xN ) =
w∈W (n,S,N )
wt(w).
(18)
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181
4.2 Standard tableaux, descents, and reading words
A tableau T with n cells is called standard iff x T = x1 x2 · · · xn . Let SYT(λ/ν) denote
the set of standard tableaux of shape λ/ν. If T is a standard tableau with n cells,
let Des(T ) be the set of all labels k < n such that k + 1 appears in a higher row
than k in T . For example, the following picture illustrates a standard tableau T ∈
SYT((4, 3, 3, 2)) with Des(T ) = {2, 5, 7, 9}.
10 11
6 8 12
3 4 7
1 2 5 9
The following equivalent description of Des(T ) will also be needed. Given a skew
shape λ/ν with n cells, totally order the cells of λ/ν by scanning the diagram row
by row from top to bottom, reading each row from left to right. Call this the reading
order of the cells of λ/ν. Given T ∈ SYT(λ/ν), the reading word rw(T ) is the list
of entries of T obtained by traversing the cells in the reading order. For example, the
standard tableau illustrated above has
rw(T ) = 10, 11, 6, 8, 12, 3, 4, 7, 1, 2, 5, 9.
One verifies immediately that Des(T ) is the set of all k < n such that k + 1 appears
earlier than k in rw(T ). Equivalently, Des(T ) is the descent set of the inverse of the
permutation rw(T ).
4.3 Standardization
There is a canonical method called standardization for converting a semistandard
tableau to a standard tableau of the same shape. Given a semistandard tableau T of
shape λ/ν, where |λ/ν| = n, we produce a standard tableau U = std(T ) by renumbering the cells of T with the integers 1, 2, . . . , n (in this order) according to the
following rules. First, smaller entries in T are relabeled before larger entries. Second,
entries in T equal to a given integer i must form a horizontal strip; the entries in this
strip are relabeled from left to right. The second rule may be equivalently stated: if
two cells in T have the same label, then the cell that occurs earlier in the reading
order is relabeled first. For example, the standardization of
5
2 4 6 7
T=
3 5 5
3 3 5
2 2 3
9
1 8 13 14
is U = std(T ) =
4 10 11 .
5 6 12
2 3 7
This simple combinatorial process has important ramifications for relating symmetric functions to quasisymmetric functions. Specifically, we have the following wellknown result for expanding skew Schur polynomials in terms of fundamental quasisymmetric polynomials [38, 39, Chap. 7].
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Theorem 9 Let λ/ν be a skew shape with n cells. For all N ≥ n,
sλ/ν (x1 , . . . , xN ) =
Ln,Des(U ) (x1 , . . . , xN ).
U ∈SYT(λ/ν)
Proof Recalling the combinatorial formulas for each side of the desired equation, we
must show that
wt(T ) =
wt(w).
T ∈SSYTN (λ/ν)
U ∈SYT(λ/ν) w∈W (n,Des(U ),N )
This equality is a consequence of the following bijection. Map each semistandard
tableau T ∈ SSYTN (λ/ν) to the pair (U, w) where U = std(T ) and w = w1 w2 · · · wn
is the multiset of labels appearing in T , sorted into weakly increasing order. For
instance, the semistandard tableau T in our previous example maps to (U, w), where
U = std(T ) is displayed above and
w = 2, 2, 2, 3, 3, 3, 3, 4, 5, 5, 5, 5, 6, 7.
It is clear that wt(w) = wt(T ). To check that w does lie in W (n, Des(U ), N ), suppose
wi = wi+1 = j . Then the labels i and i + 1 in U appear in two cells c and c that
were both labeled j in T . By definition of standardization, c must precede c in the
reading order, and therefore i + 1 appears to the right of i in rw(U ). So i ∈
/ Des(U ),
completing the verification that w ∈ W (n, Des(U ), N ). Finally, the map T → (U, w)
is a bijection, since we can recover T from U and w by replacing each entry i in the
standard tableau U by wi , for 1 ≤ i ≤ n. One checks (as above) that the condition
w ∈ W (n, Des(U ), N ) ensures that the tableau T built from any pair (U, w) must be
semistandard.
Note that Λn ∼
= ΛnN ⊆ QnN ∼
= Qn for N ≥ n, where the isomorphisms arise from
evaluation homomorphisms that are compatible for different choices of N . We can
therefore view Λn as a subspace of Qn in a canonical way, and in particular we have
the following abstract version of the previous result:
sλ/ν =
L|λ/ν|,Des(U ) .
U ∈SYT(λ/ν)
4.4 Combinatorial models for sλ/ν [A]
Next we describe combinatorial interpretations for plethystic transformations of skew
Schur functions. These interpretations generalize the usual combinatorial formula (4)
for evN (sλ/ν ) = sλ/ν (x1 , x2 , . . . , xN ) as a sum of weighted semistandard tableaux of
shape λ/ν.
To proceed, we need the concept of a combinatorial alphabet. This is a 4-tuple
A = (A, sgn, wt, <), where A is a finite set of letters (often taken to be a subset
of Z); sgn : A → {+1, −1} is a function specifying a sign for each letter in A; wt :
A → K[x1 , x2 , . . . , q, t, z, . . .] is a weight function assigning a monic monomial to
J Algebr Comb (2011) 33: 163–198
183
each letter in A; and < is a strict total ordering of A (which need not coincide with the
usual ordering of Z). Intuitively, combinatorial alphabets will aid us in interpreting
plethystic substitutions of the form
sgn(a) wt(a) ,
f
a∈A
which we abbreviate to f [wt(A)]. (Many of the results below extend to suitable infinite alphabets A. We leave this extension to the interested reader.)
Define a semistandard super-tableau of type A and shape λ/ν to be a filling T
of λ/ν by elements of A such that: (i) entries in T weakly increase (relative to the
ordering < of A) reading across rows and up columns; (ii) for each positive letter i ∈
A, the occurrences of i in T form a horizontal strip; (iii) for each negative letter j ∈ A,
the occurrences of j in T form a vertical strip. More formally, writing A = {a1 <
a2 < · · · < ap }, we can identify T with a sequence of partitions (μ0 ⊆ μ1 ⊆ · · · ⊆
μp ) such that μ0 = ν, μp = λ, μi /μi−1 is a horizontal strip whenever sgn(ai ) = +1,
and μi /μi−1 is a vertical strip whenever sgn(ai ) = −1; here μi /μi−1 consists of the
cells containing ai in T . Let SSYTA (λ/ν) denote the set of all such super-tableaux.
The signed weight of a super-tableau T is the product of the signs and weights of
all letters appearing in T . Writing T = (μ0 ⊆ · · · ⊆ μp ) as above, we have wt(T ) =
p
|μi /μi−1 | .
i=1 (sgn(ai ) wt(ai ))
Example 7 Suppose we wish to evaluate a plethystic expression of the form sλ/ν [(1 −
q)(1 − t)]. Define a combinatorial alphabet A by setting A = {a < b < c < d},
sgn(a) = sgn(d) = −1, sgn(b) = sgn(c) = +1, wt(a) = q, wt(b) = 1, wt(c) = qt,
and wt(d) = t. The following picture gives an example of a semistandard supertableaux T ∈ SSYTA ((5, 4, 4, 1)/(2, 1)).
a
a c c d
a b d
a b b
Formally, we can write
T = (2, 1), (3, 2, 1, 1), (5, 3, 1, 1), (5, 3, 3, 1), (5, 4, 4, 1) .
The signed weight of T is (−q)4 13 (qt)2 (−t)2 = +q 6 t 4 . It will follow from the next
theorem that s(5,4,4,1)/(2,1) [(1 − q)(1 − t)] is the sum of the weights all such semistandard super-tableaux of shape (5, 4, 4, 1)/(2, 1).
Theorem 10 Let A = (A, sgn, wt, <) be a combinatorial alphabet. For all partitions
ν ⊆ λ,
wt(T ).
(19)
sλ/ν [wt(A)] =
T ∈SSYTA (λ/ν)
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Proof As we will see, this result follows from the addition and negation formulas for
skew Schur functions (Sect. 3). For any plethystic alphabets A1 , . . . , Ap , iteration of
the plethystic addition formula gives
p
sλ/ν [A1 + · · · + Ap ] =
sμi /μi−1 [Ai ].
(20)
ν=μ0 ⊆μ1 ⊆···⊆μp =λ i=1
Write A = {a1 < a2 < · · · < ap }, and take Ai to be the single monomial sgn(ai )
wt(ai ). In the case sgn(ai ) = +1, it follows that
sμi /μi−1 [Ai ] = sμi /μi−1 (x1 )|x1 =wt(ai ) ,
which is wt(ai )|μ /μ | if μi /μi−1 is a horizontal strip, and zero otherwise. In the
case sgn(ai ) = −1, we get
i
i−1
sμi /μi−1 [Ai ] = (−1)|μ /μ
i
i−1 |
s(μi ) /(μi−1 ) (x1 )|x1 =wt(ai ) ,
which is (− wt(ai ))|μ /μ | if μi /μi−1 is a vertical strip, and zero otherwise. Using
these observations in (20), we see that the nonzero summands are indexed by the
semistandard super-tableaux T ∈ SSYTA (λ/ν), and the value of the summand for T
is precisely wt(T ).
i
i−1
An important remark is that the combinatorial expression appearing on the right
side of (19) depends on the total ordering < of the alphabet A, but the left side
of (19) does not depend on this total ordering (since addition in Z is commutative).
We therefore
obtain several different combinatorial interpretations for sλ/ν [wt(A)] =
sλ/ν [ a∈A sgn(a) wt(a)] by varying the total ordering imposed on A. This fact
proves to be quite useful in certain applications [23].
4.5 Plethystic calculus for quasisymmetric functions
We are now ready to define a version of plethysm that applies to quasisymmetric
functions. Let A = (A, sgn, wt, <) be a combinatorial alphabet. Write A+ = {a ∈ A :
sgn(a) = +1} and A− = {a ∈ A : sgn(a) = −1}. For n ≥ 1 and S ⊆ {1, 2, . . . , n − 1},
define W (n, S, A) to be the set of all words w = w1 w2 · · · wn such that: wk ∈ A
for all k; w1 ≤ w2 ≤ · · · ≤ wn (relative to the given ordering of A); for all k < n,
−
wk = wk+1 ∈ A+ implies k ∈
/ S; and
n for all k < n, wk = wk+1 ∈ A implies k ∈ S.
For w ∈ W (n, S, A), let wt(w) = i=1 sgn(wi ) wt(wi ). Define
LA
n,S =
wt(w).
(21)
w∈W (n,S,A)
(This definition is motivated by the analogy between (4) and (19) on the one hand,
and (18) and (21) on the other hand.) Finally, we define a linear map E A with
domain
Qn by setting E A (Ln,S ) = LA
and
extending
by
linearity.
Thus
if
f
=
S cS Ln,S
n,S
J Algebr Comb (2011) 33: 163–198
185
is any quasisymmetric function of degree n, the plethystic transform of f relative to
A is given by
E A (f ) =
c S LA
n,S .
S
Example 8 Let A = {a, b, c}, sgn(a) = sgn(b) = +1, sgn(c) = −1, wt(a) = x,
wt(b) = y, wt(c) = z. Define two total orderings on A by letting a <1 b <1 c and
c <2 b <2 a. Let A1 = (A, sgn, wt, <1 ) and A2 = (A, sgn, wt, <2 ). On one hand,
W (3, {1}, A1 ) = {abc, abb} and
2
1
LA
3,{1} = −xyz + xy .
On the other hand, W (3, {1}, A2 ) = {baa, cba, ccb, cca, cbb, caa} and
2
2
2
2
2
2
LA
3,{1} = x y − xyz + yz + xz − y z − x z.
This example shows that the value of the plethystic transformation of a quasisymmetric function can depend on the total ordering of the alphabet A.
The main result we want to prove is that this new plethysm operation on Qn agrees
with the previous notion of plethysm on the subspace Λn of symmetric functions. In
particular, upon restriction to this subspace, the answers we get do not depend on the
chosen total ordering of the alphabet. The next lemma provides the key link between
the two kinds of plethysm.
Lemma 3 Suppose A = (A, sgn, wt, <) is a combinatorial alphabet and λ/ν is a
skew shape with n cells. Then
E A (sλ/ν ) = sλ/ν wt(A) .
Proof Recall that sλ/ν = U ∈SYT(λ/ν) Ln,Des(U ) . Invoking the definition of E A and
Theorem 10, we are reduced to proving the combinatorial identity
U ∈SYT(λ/ν) w∈W (n,Des(U ),A)
wt(w) =
wt(T ).
T ∈SSYTA (λ/ν)
This follows from a standardization argument entirely analogous to the one in
Sect. 4.3. Given T ∈ SSYTA (λ/ν), we relabel the entries of T using the integers
1, 2, . . . , n via the following standardization rules.
– Rule 1: Smaller labels in T (relative to the ordering < of A) are relabeled before
larger labels.
– Rule 2: If two cells in T contain the same positive letter, then the cell that occurs
earlier in the reading order is relabeled first.
– Rule 3: If two cells in T contain the same negative letter, then the cell that occurs
later in the reading order is relabeled first.
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Using the definition of SSYTA (λ/ν), one checks that applying these rules to T produces a uniquely determined standard tableau U ∈ SYT(λ/ν). Define w = w1 · · · wn
by letting wi ∈ A be the label in T that was replaced by the label i in U . Clearly,
the passage from T to the pair (U, w) is weight-preserving. By Rule 1, we have
/ Des(U ). By Rule 3,
w1 ≤ w2 ≤ · · · ≤ wn . By Rule 2, wk = wk+1 ∈ A+ implies k ∈
wk = wk+1 ∈ A− implies k ∈ Des(U ). It follows that w ∈ W (n, Des(U ), A), as
required. Conversely, given any such pair (U, w), one recovers the tableau T ∈
SSYTA (λ/ν) by replacing the integer i in U by the letter wi for each i.
Example 9 Continuing Example 7, the tableau
a
a c c d
T=
a b d
a b b
is mapped to the pair (U, w), where U is the standard tableau
4
3 8 9 11
2 5 10
1 6 7
and w = aaaabbbccdd. The reader should confirm that Des(U ) = {1, 2, 3, 7, 10} and
w ∈ W (11, Des(U ), A).
Theorem 11 Suppose A = (A, sgn, wt, <) is a combinatorial alphabet and f ∈
Λn ⊆ Qn . Then
f wt(A) = E A (f ).
In particular, the right side does not depend on the given ordering of A.
Proof By the lemma, the two linear maps (f → f [wt(A)] : f ∈ Λn ) and (f →
E A (f ) : f ∈ Λn ) have the same effect on every Schur function. Since the Schur
functions form a basis for Λn , the two maps must be equal. The last part of the theorem follows since the ordinary plethysm f [wt(A)] does not depend on the ordering
of A.
4.6 Application: plethystic evaluation of LLT polynomials
The result in Theorem 11 leads to the following strategy for finding combinatorial
interpretations for plethystic expressions of the form f [U ], where f ∈ Λn :
(a) Define a combinatorial alphabet A such that wt(A) = U .
(b) Express f in terms of the fundamental quasisymmetric functions Ln,S .
(c) Use the expression in (b) to compute E A (f ).
Step (a) is usually easy—just write U as a sum of signed, monic monomials. In
many situations, step (b) can be accomplished by a straightforward combinatorial
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187
standardization argument. In this case, an analogous “superized” version of the same
standardization argument will often suffice to achieve step (c). We have already seen
an example of this analogy in the case f = sλ/ν (compare the proof of Theorem 9 in
Sect. 4.3 with the proof of Lemma 3 in Sect. 4.5). One technical point: we must prove
that f does lie in Λn (not merely Qn ), if this is not obvious from the definition of f .
This point is important since quasisymmetric plethysm depends on the total ordering
of the alphabet, whereas symmetric plethysm does not.
The strategy just outlined was used in [23] to give combinatorial interpretations
of plethystically transformed Macdonald polynomials. We now give another illustration of the same strategy by deriving combinatorial formulas for plethystically
transformed Lascoux–Leclerc–Thibon (LLT) polynomials [30].
First we give a combinatorial definition of the LLT polynomials. Let
Γ = λ1 /ν 1 , . . . , λs /ν s
be an ordered list of skew shapes consisting of n total squares. Let SSYTN (Γ ) denote
the set of
all lists T = (T1 , . . . , Ts ) such that Tk ∈ SSYTN (λk /ν k ) for all k. We write
wt(T) = k wt(Tk ) to keep track of which labels appear in T. Furthermore, we assign
an additional weight dinv(T) to T as follows. If c = (i, j ) is a cell in λk /ν k for any k,
we say that c belongs to the diagonal d(c) = j − i. Suppose (c1 , c2 ) is a pair of cells
in T such that c1 is labeled u and c2 is labeled v. These cells constitute a diagonal
inversion of T iff c1 ∈ λk /ν k and c2 ∈ λl /ν l for some k < l; and either d(c1 ) = d(c2 )
and u > v, or else d(c1 ) = d(c2 ) − 1 and v > u. Let dinv(T) be the total number of
diagonal inversions in T. Finally, define
LLTΓ (x1 , . . . , xN ) =
q dinv(T) wt(T) ∈ Q(q)[x1 , . . . , xN ].
T∈SSYTN (Γ )
It is true but non-obvious [23] that LLTΓ (x1 , . . . , xN ) is a symmetric polynomial in
the xi ’s; we shall not prove this fact here. Taking N ≥ n and applying ev−1
N , we obtain
an abstract symmetric function LLTΓ ∈ Λn .
Example 10 Let Γ = ((3, 2, 2, 1)/(1), (2, 2, 1), (3)). Figure 1 shows an element T ∈
SSYTN (Γ ), where N ≥ 5. It is convenient to align the tableaux diagonally, as shown
here, so that diagonal inversions are more readily computed. We have dinv(T) = 10
and wt(T ) = x13 x24 x33 x43 x52 .
Following our strategy, the next task is to use standardization to discover the
expansion of LLT polynomials in terms of fundamental quasisymmetric functions.
Given a tuple Γ with n cells total, let SYT(Γ ) be the set of all U ∈ SSYTn (Γ ) in
which the labels 1, 2, . . . , n appear once each; elements of SYT(Γ ) are called standard. Define the reading order of the cells of Γ by traversing each diagonal in turn
from southwest to northeast, working from higher diagonals to lower diagonals. Define the reading word rw(U) of a standard object to be the list of labels encountered
when the cells of U are scanned in the reading order. Define Des(U) to be the set of
all k < n such that k + 1 appears before k in rw(U).
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Fig. 1 An object T counted by
an LLT polynomial
Just as before, we can use a standardization bijection T → (U, w) to prove the
identity
q dinv(T) wt(T) =
q dinv(U)
wt(w).
(22)
T∈SSYTN (Γ )
U∈SYT(Γ )
w∈W (n,Des(U),N )
Given T, relabel the cells of T with the integers 1, 2, . . . , n such that cells with
smaller labels in T get relabeled first and the set of cells with a given label in
T are relabeled according to the reading order. This relabeling defines the standardization U = std(T); as usual, we let w be the list of all labels in T written in
weakly increasing order. It follows easily from the definitions of dinv and standardization that w ∈ W (n, Des(U), N ) and dinv(U) = dinv(T); furthermore, it is clear
that wt(w) = wt(T). So (22) holds. Recalling the relevant definitions and applying
ev−1
N , we obtain the desired abstract expansion
LLTΓ =
q dinv(U) Ln,Des(U) .
(23)
U∈SYT(Γ )
Example 11 The object T shown in Fig. 1 standardizes to give the standard object U
shown in Fig. 2. In this case, w = 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5;
rw(U) = 14, 8, 11, 4, 12, 9, 5, 1, 13, 6, 2, 10, 7, 3, 15;
and Des(U) = {3, 7, 10, 13}. The reader should confirm that dinv(U) = dinv(T) and
that w ∈ W (15, Des(U), 5).
We can now compute plethysms of the form LLTΓ [Z]. Let A = (A, sgn, wt, <)
be a combinatorial alphabet such that wt(A) = Z. Define SSYTA (Γ ) to be the set
of all tuples T = (T1 , . . . , Ts ) such that Tk ∈ SSYTA (λk /ν k ) for all k. To standardize
such an object, relabel the cells in T with the integers 1, 2, . . . , n so that earlier letters
of A (relative to <) are relabeled first, equal positive letters of A are relabeled in the
reading order, equal negative letters of A are relabeled in reverse reading order. The
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189
Fig. 2 Standardization of T
resulting object U = std(T) is easily seen to be standard. Furthermore, letting w =
w1 w2 · · · wn be the list of labels in T in weakly increasing order, the standardization
rules show immediately that w ∈ W (n, Des(U), A). We conclude that
q dinv(T) wt(T) =
T∈SSYTA (Γ )
q dinv(U)
wt(w),
w∈W (n,Des(U),A)
U∈SYT(Γ )
where (by definition) dinv(T) = dinv(std(T)) and wt(T) is the product of the signs
and weights of all entries of T. Comparing this expansion to (23), and recalling the
formula for LA
n,Des(U) , we have proved:
LLTΓ [Z] = LLTΓ wt(A) =
q dinv(T) wt(T).
T∈SSYTA (Γ )
5 Plethystic calculus and dual bases
Some of the most useful plethystic identities are Cauchy formulas for simplifying
expressions of the form hn [AB] or en [AB] [28, pp. 38, 43]. Before proving these, we
review the non-plethystic versions of the Cauchy identities, in which dual bases of Λ
play a key role.
5.1 Review of dual bases
We define the Hall inner product on the vector space Λ by setting pλ , pμ = δλ,μ zμ
and extending by bilinearity. Here, δλ,μ is 1 for λ = μ and 0 otherwise. Relative to
this inner product, the basis {pμ : μ ∈ Par(n)} of Λn is dual to the basis {pμ /zμ : μ ∈
Par(n)}, whereas the basis {sμ : μ ∈ Par(n)} is self-dual (orthonormal). Furthermore,
the hμ ’s are dual to the monomial basis mμ , and the eμ ’s are dual to the forgotten
basis fμ . One easily sees that the involution ω is an isometry relative to this scalar
product: ω(f ), ω(g) = f, g for all f, g ∈ Λ.
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Suppose B = {Bμ : μ ∈ Par(n)} and C = {Cμ : μ ∈ Par(n)} are two bases of Λn .
It can be shown that B and C are dual bases (relative to the Hall scalar product) iff
the following Cauchy identity holds:
Bμ (x1 , . . . , xM )Cμ (y1 , . . . , yN ) =
M N
i=1 j =1
μ∈Par(n)
1
1 − xi yj terms of degree 2n
(M, N ≥ n).
We note that the right side of this identity can also be written
hn (z1 , z2 , . . . , zMN )|z1 →x1 y1 ,...,zMN →xM yN ;
here we are using the UMP for the polynomial ring K[z1 , . . . , zMN ]. There is also a
dual Cauchy identity, which states that B and C are dual bases iff
Bμ (x1 , . . . , xM ) ω(Cμ ) (y1 , . . . , yN )
μ∈Par(n)
=
M N
(1 + xi yj )
i=1 j =1
(M, N ≥ n)
terms of degree 2n
= en (z1 , . . . , zMN )|z1 →x1 y1 ,...,zMN →xM yN .
For many choices of the bases Bμ and Cμ , these polynomial identities have nice
combinatorial or algebraic proofs. For example, when Bμ = Cμ = sμ , the Cauchy
identities follow from the RSK algorithm [37–39].
5.2 Plethystic Cauchy identities
As before, we find it convenient to phrase the “plethystic Cauchy identities” in a
slightly more general form involving ring homomorphisms. Let S be a K-algebra
and D, E two K-algebra homomorphisms of Λ into S. By the UMP, there is a unique
K-algebra homomorphism D ∗P E of Λ into S sending pk to D(pk )E(pk ) for all
k ≥ 1.
Theorem 12 If B = {Bμ : μ ∈ Par(n)} and C = {Cμ : μ ∈ Par(n)} are dual bases for
Λn , then
D(Bμ )E(Cμ ),
(24)
(D ∗P E)(hn ) =
μ∈Par(n)
(D ∗P E)(en ) =
D(Bμ )E ω(Cμ ) .
(25)
μ∈Par(n)
Proof We adapt the usual proof of the classical Cauchy identities [34, Sect. I.4] to
incorporate the homomorphisms D and E. There exist unique scalars bμ,ν , cμ,ν ∈ K
J Algebr Comb (2011) 33: 163–198
so that
Bμ =
191
Cμ =
bμ,ν pν ,
ν∈Par(n)
cμ,ν (pν /zν ).
ν∈Par(n)
Let U = (bμ,ν )μ,ν∈Par(n) and V = (cμ,ν )μ,ν∈Par(n) denote matrices obtained from
these scalars by arranging the partitions of n in any fixed order. Since B and C
are dual bases, a short calculation with inner products reveals that U V T = I . This
condition is equivalent to V T U = I and hence to U T V = I . We therefore have
ν, ξ ∈ Par(n) .
bμ,ν cμ,ξ = δν,ξ
μ∈Par(n)
Now, using K-linearity of D and E, compute
D(Bμ )E(Cμ ) =
D
bμ,ν pν E
cμ,ξ pξ /zξ
μ∈Par(n)
μ∈Par(n)
=
ξ ∈Par(n)
ν∈Par(n)
D(pν )E(pξ ) bμ,ν cμ,ξ
zξ
ν∈Par(n) ξ ∈Par(n)
μ∈Par(n)
(D ∗P E)(pν )
D(pν )E(pν )
=
.
=
zν
zν
ν∈Par(n)
ν∈Par(n)
Since hn = ν pν /zν (as follows from (6)), we see that the last expression is just
(D ∗P E)(hn ).
To deduce (25) from (24), we first observe the identity
(D ∗P E) ◦ ω = D ∗P (E ◦ ω) = (D ◦ ω) ∗P E.
(26)
This follows by the universal mapping property, since all three homomorphisms send
pk to (−1)k−1 D(pk )E(pk ) ∈ S. Now compute
(D ∗P E)(en ) = (D ∗P E) ◦ ω (hn ) = D ∗P (E ◦ ω) (hn )
=
D(Bμ )E ω(Cμ ) ,
μ∈Par(n)
where the last step is an application of (24) to the homomorphisms D and E ◦ ω. Some useful particular cases of formulas (24) and (25), written in plethystic notation, are as follows:
hn [AB] =
pμ [A]pμ [B]/zμ =
hμ [A]mμ [B] =
mμ [A]hμ [B]
μ∈Par(n)
=
μ∈Par(n)
en [AB] =
μ∈Par(n)
μ∈Par(n)
eμ [A]fμ [B] =
μ∈Par(n)
fμ [A]eμ [B] =
μ∈Par(n)
sμ [A]sμ [B],
μ∈Par(n)
(−1)n−(μ) pμ [A]pμ [B]
=
hμ [A]fμ [B]
zμ
μ∈Par(n)
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=
mμ [A]eμ [B]
μ∈Par(n)
=
eμ [A]mμ [B] =
μ∈Par(n)
fμ [A]hμ [B] =
μ∈Par(n)
sμ [A]sμ [B].
μ∈Par(n)
One can prove a converse to Theorem 12, in which the duality of the bases B and
C can be deduced if (24) holds for sufficiently many homomorphisms D and E. We
will not give a precise statement or proof of this converse, since it will not be needed
in the sequel.
5.3 The Ω operator
In [16], Garsia et al. encode various symmetric function “kernels” via plethystic substitutions of the form Ω[A], where
Ω=
i≥1
∞
1
=
hn = exp
pk /k .
1 − xi
n=0
k≥1
Some care must be exercised here
since, as remarked in [34, Chap. I, Sect. 2], Ω is
not an element of the ring Λ = n≥0 Λn . Rather, Ω is an element of the larger ring
Λ̂, which is the F -module n≥0 Λn endowed with the natural multiplication.
To make sense of the symbol Ω[A], we need to assume that the associated homomorphism φA : Λ → S takes its values in a topological ring S, e.g., a ring of formal
power series. Then we define
Ω[A] = lim
N
N →∞
hn [A] = lim
N →∞
n=0
N
φA (hn ) ∈ S,
n=0
provided that this limit exists in S. (See [7] for a discussion of limits in rings
of formal power series.) For example, consider the alphabet A = 0. We have
φA (h0 ) = φA (1) = 1 and φA (hn ) = 0 for all n > 0. Therefore, Ω[0] = 1.
The most important property of Ω is that it converts sums to products:
Ω[A + B] = Ω[A]Ω[B]
for all alphabets A and B such that both sides are defined. To prove this, use the
addition formula to compute
Ω[A + B] =
hn [A + B] =
n≥0
=
n
hk [A]hj [B]
k≥0 j ≥0
=
k≥0
hk [A]hn−k [B]
n≥0 k=0
hk [A] ·
hj [B] = Ω[A]Ω[B].
j ≥0
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193
Setting B = −A, we deduce the negation formula
Ω[−A] = 1/Ω[A].
Suppose m is a monic monomial in S unequal to 1. Using Theorem 7, we find that
Ω[m] =
hn [m] =
mn = 1/(1 − m) ∈ S.
n≥0
n≥0
Ω[txi yj ] =
1
,
1 − txi yj
For example,
Ω[xi yj ] =
1
,
1 − xi yj
Ω[q k xi yj ] =
1
.
1 − q k xi yj
Using the addition formula for Ω, we immediately deduce the following finite product expansions (where Xn = x1 + · · · + xn and Ym = y1 + · · · + ym ):
Ω[Xn Ym ] =
n m
i=1 j =1
n m
1 − txi yj
Ω Xn Ym (1 − t) =
.
1 − xi yj
1
,
1 − xi yj
i=1 j =1
To extend these formulas to infinite products, we need one more limiting operation. Suppose A = n≥1 An is an “infinite sum of alphabets.” We define
Ω[A] = lim Ω[A1 + · · · + AN ]
N →∞
provided that this limit exists in S. In this situation, we have Ω[A] =
since
Ω[A] = lim Ω[A1 + · · · + AN ] = lim
N →∞
N →∞
N
Ω[An ] =
n=1
∞
n≥1 Ω[An ],
Ω[An ].
n=1
Let X = x1 + x2 + · · · = p1 ⊗ 1 ∈ Λ ⊗K Λ and Y = y1 + y2 + · · · = 1 ⊗ p1 ∈ Λ ⊗K Λ.
Then
Ω[XY ] =
∞ ∞
i=1 j =1
1
,
1 − xi yj
∞ ∞
1 − txi yj
Ω XY (1 − t) =
,
1 − xi yj
i=1 j =1
Ω XY
∞ ∞ ∞
1 − tq k xi yj
1−t
=
.
1−q
1 − q k xi yj
i=1 j =1 k=0
The last formula follows by writing
for Ω.
1
1−q
=
k≥0 q
k
and using the addition rule
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J Algebr Comb (2011) 33: 163–198
5.4 Application to operators in Macdonald theory
We end this section by giving an example of how plethystic notation can be used in
the theory of Macdonald polynomials. Macdonald [33] introduced an operator δ1 that
is defined as follows. Given a polynomial P (x1 , . . . , xn ), let
Tq(s) P (x1 , . . . , xn ) = P (x1 , . . . , xs−1 , qxs , xs+1 , . . . , xn )
(27)
and
δ1 P (x1 , . . . , xn ) =
n n
txs − xi
s=1 i=1
i=s
=
xs − xi
Tq(s) P (x1 , . . . , xn )
n
(s)
Tt Δ(x1 , . . . , xn )
s=1
Δ(x1 , . . . , xn )
Tq(s) P (x1 , . . . , xn )
(28)
where Δ(x1 , . . . , xn ) = 1≤i<j ≤n (xi − xj ) is the Vandermonde determinant. Macdonald proved the following facts about the
operator δ1 . Given a composition p =
(p1 , . . . , pn ) with n parts, let ωp (q, t) = ni=1 t n−i q pi . Recall that p + is the partition obtained by sorting the parts of p into decreasing order. Macdonald proved that
for all partitions λ of n,
mμ
δ1 mλ (x1 , . . . , xn ) = ωλ (q, t)mλ (x1 , . . . , xn )+
μ<D λ
εμ (p)ωp (q, t),
p=(p1 ,...,pn ):
p + =λ
(29)
where εμ (p) = sgn(σ ) if there is a permutation σ such that pσi + n − i = μi + n − i
for i = 1, . . . , n and εμ (p) = 0 if there is no such σ , and <D denotes the dominance
ordering on partitions. It immediately follows that for all partitions λ of n,
δ1 sλ (x1 , . . . , xn ) = ωλ (q, t)sλ (x1 , . . . , xn ) +
dλ,μ (q, t)sμ (x1 , . . . , xn ) (30)
μ<D λ
for some polynomials dλ,μ (q, t). In fact, Macdonald proved that if
zλ (x) =
μ∈Par(n)−{λ}
x − ωμ (q, t)
,
ωλ (q, t) − ωμ (q, t)
(31)
then the Macdonald polynomial Pλ (x1 , . . . , xn ; q, t) is given by
Pλ (x1 , . . . , xn ; q, t) = zλ (δ1 )sλ (x1 , . . . , xn ).
(32)
Garsia and Haiman [15] gave a plethystic interpretation for the operator δ1 which
they used to prove a number of fundamental results about the q, t-Catalan numbers,
and which was later used by Remmel [36] to develop the combinatorics of δ1 applied
J Algebr Comb (2011) 33: 163–198
195
to Schur functions. That is, letting Xn = x1 + · · · + xn , Garsia and Haiman proved
that for any symmetric polynomial P ,
P [Xn ]
q −1
tn
Ω z(t − 1)Xn .
δ1 P [Xn ] =
(33)
+
P Xn +
1−t
t −1
tz
z0
(i)
More generally, note that in plethystic notation Tq (P [Xn ]) = P [Xn + (q − 1)xi ].
Garsia [12] introduced the following family of Macdonald-like operators
D1(k) =
n n
txi − xj k (i)
x T
xi − xj i q
(34)
i=1 j =1
j =i
and proved that for any symmetric polynomial P ,
(k) D1 P [Xn ] = χ(k
t n−k
P [Xn ]
q −1
+
P Xn +
Ω z(t − 1)Xn . (35)
= 0)
1−t
t −1
tz
zk
We now present Garsia’s proof [12] of (35) because it shows the remarkable power
of the plethystic calculus. Both sides of (35) are linear in P , so it suffices to verify the
formula when P is a Schur symmetric function. Letting Y = u(y1 + · · · + yn + · · · ) =
1 ⊗ p1 u, we have
Ω[Xn Y ] =
hN [Xn Y ] =
N ≥0
=
N ≥0
uN
sλ [Xn ]sλ [Y ]
N ≥0 λ∈Par(N )
sλ (x1 , . . . , xn ) ⊗ sλ ∈ (Λn ⊗K Λ) [u] .
(36)
λ∈Par(N )
Using Ω[Xn Y ] as the generating function for the Schur polynomials sλ [Xn ], we can
establish (35) for all Schur functions by showing that
(k) D1
t n−k
Ω[Xn Y ]
+
Ω
Ω[Xn Y ] = χ(k = 0)
1−t
t −1
× Ω z(t − 1)Xn Xn +
q −1
Y
tz
zk
t n−k
Ω[Xn Y ]
q −1
+
Ω[Xn Y ]Ω
Y
1−t
t −1
tz
× Ω z(t − 1)Xn .
= χ(k = 0)
(37)
zk
(More precisely, the left side is the image of Ω[Xn Y ] under the linear map on
(k)
(Λn ⊗K Λ)[[u]] that applies D1 ⊗ 1 to each coefficient of uN ; the desired result for
Schur polynomials will follow from (37) by taking the coefficient of uN and using the
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J Algebr Comb (2011) 33: 163–198
(i)
linear independence of {1 ⊗ sλ : λ ∈ Par(N )}.) Recalling that Tq (P [Xn ]) = P [Xn +
(k)
(q − 1)xi ], we see from the definition of D1 that
(k) D1 Ω[Xn Y ] =
n n
txi − xj k x Ω Xn + (q − 1)xi Y
xi − xj i
i=1 j =1
j =i
or, equivalently,
D1 (Ω[Xn Y ]) txi − xj k xi Ω (q − 1)xi Y .
=
Ω[Xn Y ]
xi − xj
n
(k)
n
(38)
i=1 j =1
j =i
Next, the definition of Ω gives
Ω (q − 1)xi Y =
hm (q − 1)Y xim .
(39)
m≥0
Substituting (39) into (38), we obtain
txi − xj k+m
D1 (Ω[Xn Y ]) =
hm (q − 1)Y
x
.
Ω[Xn Y ]
xi − xj i
n
(k)
n
(40)
i=1 j =1
j =i
m≥0
Observe that Ω[(t − 1)Xn z] =
form
n
1−zxi
i=1 1−ztxi
has a partial fraction expansion of the
n
n 1 1
Ω (t − 1)Xn z = n +
Ai (x, t)
= t −n +
Ai (x, t)zm t m xim . (41)
t
1 − ztxi
i=1
i=1 m≥0
Fixing i, multiplying both sides of (41) by 1 − ztxi , and taking the limit as z →
gives
Ai (x, t) =
n
t − 1 txi − xj
.
tn
xi − xj
1
txi
(42)
j =1
j =i
Substituting (42) into (41) and taking the coefficient of zm on both sides, we obtain
n n
txi − xj m
1
1 t n Ω[(t − 1)Xn z]
.
−
xi = m
xi − xj
t
t −1
t − 1 z m
i=1 j =1
j =i
One can then use (43) with m replaced by m + k to show that (40) becomes
n
(k)
1
D1 (Ω[Xn Y ]) t Ω[(t − 1)Xn z]
1
=
−
hm (q − 1)Y m+k
Ω[Xn Y ]
t
t −1
t − 1 zm+k
m≥0
(43)
J Algebr Comb (2011) 33: 163–198
197
n
(q − 1)Y
t Ω[(t − 1)Xn z]
1
1
(zt)m m+k
−
tz
t
t −1
t − 1 zm+k
m≥0
(q − 1)Y 1 t n Ω[(t − 1)Xn z]
1
.
=
hm
(44)
−
k
tz
t
t −1
t − 1 z k
=
hm
m≥0
Observing that the term
if k > 0, we see that
1
t−1
inside the big parentheses does not contribute anything
(k)
D1 (Ω[Xn Y ])
1
t n−k (q − 1)
= χ(k = 0)
+
Ω (t − 1)Xn z
Y ] hm [
Ω[Xn Y ]
1−t t −1
tz
zk
m≥0
(q − 1) t n−k 1
+
Ω (t − 1)Xn z Ω
Y .
= χ(k = 0)
1−t t −1
tz
zk
(45)
Multiplying both sides of (45) by Ω[Xn Y ] gives (37), which is what we needed to
prove.
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